Welcome to my Set Theory Page

Set Theory: Foundations of Mathematics

The two axioms which define Intuitive Set Theory are discussed below.

  • Ackermann Functions and Transfinite Ordinals
    An important part of Cantor's set theory, which forms the foundations of mathematics, is the concept of transfinite ordinals. A systematic way of writing the sequence of ordinals is given.
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  • Justification of the Continuum Hypothesis
    Intuitive arguments are given to suggest that Continuum Hypothesis should be accepted.
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  • Axiomatic Derivation of the Continuum Hypothesis
    Continuum Hypothesis is derived from an axiom called Axiom of Monotonicity.
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  • Real Set Theory
    A set theory is defined in which Generalized Continuum Hypothesis and Axiom of Choice are theorems.
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  • Intuitive Set Theory
    A set theory is defined in which Skolem Paradox does not arise. Also, there are no sets which are not Lebesgue measurable.
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  • Two Axioms to Extend Zermelo-Fraenkel Theory
    Axiom of Monotonicity is used along with Zermelo-Fraenkel set theory to derive Generalized Continuum Hypothesis. Axiom of Fusion is used to investigate the cardinality of the set of points in a unit interval.
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  • Visualization of Intuitive Set Theory
    Intuitive Set Theory is defined as the theory we get when axioms of Monotonicity and Fusion are added to ZF theory. Cardinals in the theory are visualized using illustrations.
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  • White Hole, Black Whole, and The Book
    Physical and intellectual spaces are visualized making use of concepts from Intuitive Set Theory. A book containing all the proofs of mathematics is called The Book.
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  • The Essence of Intuitive Set Theory
    The ideas which motivated the defintion of intuitive set theory are explained. Only a passing acquaintance with the transfinite cardinals of Cantor is assumed on the part of the reader.
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  • Generalized Continuum Hypothesis and the Method of Fusing
    The method of fusing explains the basis for the formulation of the axioms of monotonicity and fusion, the two axioms which define intuitive set theory.
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  • Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets
    Axiom of Combinatorial Sets is defined and used to derive generalized continuum hypothesis.
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  • Definition of Intuitive Set Theory
    Two axioms which define intuitive set theory, Axiom of Combinatorial Sets and Axiom of Infinitesimals, are stated. Generalized continuum hypothesis is derived from the first axiom, and the infinitesimal is visualized using the latter axiom.
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  • Two Open Problems and a Conjecture in Mathematical Logic
    The open problems attempt to extend Zermelo-Fraenkel set theory and the conjecture suggests an extension of Godel's incompleteness theorems.
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  • Teaching Generalized Continuum Hypothesis
    Generalized Continuum Hypothesis is derived from a simple axiom called Axiom of Combinatorial Sets.
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  • The Mathematical Universe in a Nutshell
    The mathematical universe discussed here gives models of possible structures our physical universe can have.
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  • Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets
    Continuum Hypothesis is derived from an axiom called Axiom of Combinatorial Sets. The derivation is simple enough to be understood by any novice, who has a passing acquintance with cardinals of Cantor.
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The advantage with the intuitive set theory is that it allows us to have a simple visualization of our physical and intellectual spaces.

K. K. Nambiar

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